Kantorovich Dual Problem (for general costs): Difference between revisions
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(Kantorovich Duality) Let X and Y be Polish spaces, let <math>\mu \in \mathcal{P}(X)</math> and <math>\nu \in \mathcal{P}(Y)</math>, and let a cost function <math> c:X \times Y \rightarrow[0,+\infty] </math> be lower semi-continuous. | (Kantorovich Duality) Let X and Y be Polish spaces, let <math>\mu \in \mathcal{P}(X)</math> and <math>\nu \in \mathcal{P}(Y)</math>, and let a cost function <math> c:X \times Y \rightarrow[0,+\infty] </math> be lower semi-continuous. | ||
Whenever <math> \pi \in \mathcal{P}(X \times Y) </math> and <math> (\varphi, \psi) \in L^{1}(d\mu) \times L^{1}(d\nu) </math>, define \newline | Whenever <math> \pi \in \mathcal{P}(X \times Y) </math> and <math> (\varphi, \psi) \in L^{1}(d\mu) \times L^{1}(d\nu) </math>, define | ||
\newline | |||
<math> I[\pi]= \int_{X\times Y} c(x,y) d\pi(x,y), \quad J(\varphi,\psi)=\int_{X}\varphi(x)d\mu(x)+\int_{Y}\psi(y) d\nu(y) </math>. | <math> I[\pi]= \int_{X\times Y} c(x,y) d\pi(x,y), \quad J(\varphi,\psi)=\int_{X}\varphi(x)d\mu(x)+\int_{Y}\psi(y) d\nu(y) </math>. | ||
Revision as of 22:37, 16 May 2020
Introduction
The main advantage of Kantorovich Problem, in comparison to Monge problem, is in the convex constraint property. It is possible to formulate the dual problem.
Statement of Theorem
(Kantorovich Duality) Let X and Y be Polish spaces, let and , and let a cost function be lower semi-continuous. Whenever and , define \newline .
![{\displaystyle c:X\times Y\rightarrow [0,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86eb4eb62728fa63010b95eb2c285d7d6a18649)
![{\displaystyle I[\pi ]=\int _{X\times Y}c(x,y)d\pi (x,y),\quad J(\varphi ,\psi )=\int _{X}\varphi (x)d\mu (x)+\int _{Y}\psi (y)d\nu (y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe572f66aca1d528fc4ced0547d7fc298e00b502)
Proof of Theorem
References
</ references>
- ↑ C. Villani, Topics in Optimal Transportation, Chapter 1. (pages 17-21)
- ↑ https://link-springer-com.proxy.library.ucsb.edu:9443/book/10.1007/978-3-319-20828-2 F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1.] (pages 9-16)